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Lesson 3-9
The Pythagorean Theorem
Lesson Objectives
1 Solve problems using the
Pythagorean Theorem
2 Identify right triangles
NAEP 2005 Strand: Geometry
Topic: Relationships Among Geometric Figures
Local Standards: ____________________________________
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Vocabulary and Key Concepts.
The Pythagorean Theorem
In any right triangle, the sum of the squares of the lengths of the legs
is equal to the square of the length of the hypotenuse.
2
2
c
a
2
b
The Converse of the Pythagorean Theorem
If a triangle has sides of lengths a, b, and c, and a2 b2 c2, then the triangle is
a
with hypotenuse of length c.
The hypotenuse is
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The legs are
A conditional is
A hypothesis is
A conclusion is
A converse is
Examples.
1 Using the Pythagorean Theorem What is the length of the
hypotenuse of this triangle?
a2 b2 c2
2
2
c2
2
c
c2
c
Substitute
for a and
c
8m
Use the Pythagorean Theorem.
for b.
15 m
Simplify.
Find the principal square root of each side.
Simplify.
The length of the hypotenuse is 17 m.
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Algebra 1 Lesson 3-9
65
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2 Using the Converse of the Pythagorean Theorem Determine whether the
given lengths are sides of a right triangle.
a. 5 in., 5 in., and 7 in.
2
2
2
Determine whether a2 b2 c 2, where c is the longest side.
Simplify.
This triangle
a right triangle.
2
2
Determine whether a2 b2 c2, where c is the longest side.
Simplify.
This triangle
a right triangle.
3 Physics If two forces pull at right angles to each other, the resultant force is
Force A
eC
Force B
2
2
2
2 Determine whether a b c ,
rc
Fo
2
nt
ta
2
ul
For a 50-lb force and a 120-lb force, the resultant force is 130 lb. Are the forces
pulling at right angles to each other?
s
Re
represented as the diagonal of a rectangle, as shown in the diagram. The diagonal
forms a right triangle with two of the perpendicular sides of the rectangle.
where c is the greatest force.
The forces of 50 lb and 120 lb
at right angles to each other.
Quick Check.
1. What is the length of the hypotenuse of a right triangle with legs of lengths 7 cm
and 24 cm?
2. A triangle has sides of lengths 10 m, 24 m, and 26 m. Is the triangle a right triangle?
3. Physics Consider Example 3. For a 70-lb force and a 60-lb force, the resultant
force is 100 lb. Are the forces pulling at right angles to each other?
66
Algebra 1 Lesson 3-9
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b. 10 cm, 24 cm, and 26 cm